I recently graduated in May. I'm willing to take the next two years off to do something like take on a paralegal position. I've looked through a few postings, but many of the paralegal job postings (especially at big firms) ask for 3-5 years of experience and salary history. The thing is, I don't have any real job history to report, nor any relevant work experience. How should I go about becoming a paralegal, then?

You want to look for positions called things like Jr. Paralegal, Legal Assistant, Legal Associate, etc. Firms and large legal nonprofits all have positions specifically for people fresh out of college, and they'll be called something like what I listed.

There's a difference between the "paralegal" position you're looking for, and the "paralegal" position that goes to people who want to be career paralegals and get paralegal certificates/degrees to do so.

ETA: Unfortunately for you, a lot of these positions are filled by college seniors in March through June of their senior years, with a start date sometime in the summer after graduation. You can still find this type of two-year job, but they might be a little more scarce until they start to be posted again in Jan/Feb.

You could also consider signing up with a temp agency, which can place new paralegals with law offices on a purely temporary or a potential temporary to perm basis. BTW Many firms use the title "project assistant" to refer to junior paralegals who work there with no prior experience.

That was how I got into the field and I've been a paralegal for 8 years.